Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

DOI

10.1080/14029251.2015.1079428How to use a DOI?

Keywords

initial boundary value problems; oscillatory initial data; nonlinear Schrödinger equation

Abstract

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(−2iβx) (or asymptotically periodic, u(x, 0) =α exp(−2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: u_x(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.

Copyright

© 2015 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Spyridon Kamvissis
AU  - Dmitry Shepelsky
AU  - Lech Zielinski
PY  - 2021
DA  - 2021/01/06
TI  - Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 448
EP  - 473
VL  - 22
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2015.1079428
DO  - 10.1080/14029251.2015.1079428
ID  - Kamvissis2021
ER  -